3.7.49 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) [649]

Optimal. Leaf size=95 \[ -\frac {(1+a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {2 (1+a x)^2}{a c \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \text {ArcSin}(a x)}{a c} \]

[Out]

-1/3*(a*x+1)^3/a/c/(-a^2*x^2+1)^(3/2)-3*arcsin(a*x)/a/c+2*(a*x+1)^2/a/c/(-a^2*x^2+1)^(1/2)+3*(-a^2*x^2+1)^(1/2
)/a/c

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6292, 6283, 1649, 21, 683, 655, 222} \begin {gather*} -\frac {(a x+1)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {2 (a x+1)^2}{a c \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \text {ArcSin}(a x)}{a c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2)),x]

[Out]

-1/3*(1 + a*x)^3/(a*c*(1 - a^2*x^2)^(3/2)) + (2*(1 + a*x)^2)/(a*c*Sqrt[1 - a^2*x^2]) + (3*Sqrt[1 - a^2*x^2])/(
a*c) - (3*ArcSin[a*x])/(a*c)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 683

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(p + 1))), x] - Dist[e^2*((m + p)/(c*(p + 1))), Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(-d)*f*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2
*a*e*(p + 1))), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)
*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p
+ 1/2, 0] && GtQ[m, 0]

Rule 6283

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 6292

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u/x^(2*p))*(1
- a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {a^2 \int \frac {e^{3 \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \int \frac {x^2 (1+a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=-\frac {(1+a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {a^2 \int \frac {\left (\frac {3}{a^2}+\frac {3 x}{a}\right ) (1+a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {(1+a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {(1+a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {2 (1+a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {3 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {(1+a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {2 (1+a x)^2}{a c \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {(1+a x)^3}{3 a c \left (1-a^2 x^2\right )^{3/2}}+\frac {2 (1+a x)^2}{a c \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{a c}-\frac {3 \sin ^{-1}(a x)}{a c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 78, normalized size = 0.82 \begin {gather*} \frac {-14+5 a x+16 a^2 x^2-3 a^3 x^3-9 (-1+a x) \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{3 a c (-1+a x) \sqrt {1-a^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2)),x]

[Out]

(-14 + 5*a*x + 16*a^2*x^2 - 3*a^3*x^3 - 9*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a*c*(-1 + a*x)*Sqrt[1 -
a^2*x^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(87)=174\).
time = 0.75, size = 209, normalized size = 2.20

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2}}{c}\)	\(152\)
default	\(\frac {a^{2} \left (\frac {7 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {4}{a^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {4}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {4 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{3}}\right )}{c}\)	\(209\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x,method=_RETURNVERBOSE)

[Out]

1/c*a^2*(7*x/a^2/(-a^2*x^2+1)^(1/2)+4/a^3/(-a^2*x^2+1)^(1/2)-3/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+
1)^(1/2))+a*(-x^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-a^2*x^2+1)^(1/2))+4/a^3*(1/3/a/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x
-1/a))^(1/2)+1/3/a*(-2*a^2*(x-1/a)-2*a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))), x)

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 101, normalized size = 1.06 \begin {gather*} \frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x, algorithm="fricas")

[Out]

1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a
*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^3*c*x^2 - 2*a^2*c*x + a*c)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{2} \left (\int \frac {x^{2}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {2 a x^{3}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{2} x^{4}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2),x)

[Out]

a**2*(Integral(x**2/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 +
1) - sqrt(-a**2*x**2 + 1)), x) + Integral(2*a*x**3/(-a**3*x**3*sqrt(-a**2*x**2 + 1) + a**2*x**2*sqrt(-a**2*x**
2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a**2*x**4/(-a**3*x**3*sqrt(-a**2*x**2
+ 1) + a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 129, normalized size = 1.36 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}-\frac {2\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^4\,x^2-2\,c\,a^3\,x+c\,a^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)^3/((c - c/(a^2*x^2))*(1 - a^2*x^2)^(3/2)),x)

[Out]

(1 - a^2*x^2)^(1/2)/(a*c) - (13*(1 - a^2*x^2)^(1/2))/(3*((c*(-a^2)^(1/2))/a - c*x*(-a^2)^(1/2))*(-a^2)^(1/2))
- (3*asinh(x*(-a^2)^(1/2)))/(c*(-a^2)^(1/2)) - (2*a*(1 - a^2*x^2)^(1/2))/(3*(a^2*c + a^4*c*x^2 - 2*a^3*c*x))

________________________________________________________________________________________